Machine learning comes in two varieties:
\[\begin{equation} C(f(X_i|\theta), \;y_{i}^\text{actual}) \geq 0. \end{equation}\]
\[\begin{equation} \theta = \underset{\theta}{\operatorname{argmin}} \sum C(f(X_i|\theta), \;y_{i}^\text{actual}) + \lambda \left\lVert\theta\right\rVert, \end{equation}\]
where \(\lambda\) is some penalty for overfitting (more tomorrow)
\(\implies\) use language of probability
\[\begin{equation} p(X|\text{camera quality, location, cat genetics etc.)}. \end{equation}\]
\[\begin{equation} p(X, \text{camera quality, location, cat genetics etc.}) \end{equation}\]
\[\begin{equation} \widehat p_\theta(y|Z) \approx p(y|X), \end{equation}\]
where \(Z\) are features of \(X\)
\[\begin{equation} \widehat p_\theta(Z) \approx p(X), \end{equation}\]
where \(Z\) are features of \(X\)
| country | designation | points | price | province | region_1 | region_2 | variety | winery |
|---|---|---|---|---|---|---|---|---|
| US | Martha’s Vineyard | 96 | 235 | California | Napa Valley | Napa | Cabernet Sauvignon | Heitz |
| Spain | Carodorum Selección Especial Reserva | 96 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | |
| US | Special Selected Late Harvest | 96 | 90 | California | Knights Valley | Sonoma | Sauvignon Blanc | Macauley |
| US | Reserve | 96 | 65 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi |
| France | La Brûlade | 95 | 66 | Provence | Bandol | Provence red blend | Domaine de la Bégude | |
| Spain | Numanthia | 95 | 73 | Northern Spain | Toro | Tinta de Toro | Numanthia | |
| Spain | San Román | 95 | 65 | Northern Spain | Toro | Tinta de Toro | Maurodos | |
| Spain | Carodorum Único Crianza | 95 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | |
| US | Silice | 95 | 65 | Oregon | Chehalem Mountains | Willamette Valley | Pinot Noir | Bergström |
| US | Gap’s Crown Vineyard | 95 | 60 | California | Sonoma Coast | Sonoma | Pinot Noir | Blue Farm |
| Italy | Ronco della Chiesa | 95 | 80 | Northeastern Italy | Collio | Friulano | Borgo del Tiglio | |
| US | Estate Vineyard Wadensvil Block | 95 | 48 | Oregon | Ribbon Ridge | Willamette Valley | Pinot Noir | Patricia Green Cellars |
| US | Weber Vineyard | 95 | 48 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Patricia Green Cellars |
| France | Château Montus Prestige | 95 | 90 | Southwest France | Madiran | Tannat | Vignobles Brumont | |
| US | Grace Vineyard | 95 | 185 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Domaine Serene |
| US | Sigrid | 95 | 90 | Oregon | Willamette Valley | Willamette Valley | Chardonnay | Bergström |
| US | Rainin Vineyard | 95 | 325 | California | Diamond Mountain District | Napa | Cabernet Sauvignon | Hall |
| Spain | 6 Años Reserva Premium | 95 | 80 | Northern Spain | Ribera del Duero | Tempranillo | Valduero | |
| France | Le Pigeonnier | 95 | 290 | Southwest France | Cahors | Malbec | Château Lagrézette | |
| US | Gap’s Crown Vineyard | 95 | 75 | California | Sonoma Coast | Sonoma | Pinot Noir | Gary Farrell |
| US | Grignolino | 95 | 24 | California | Napa Valley | Napa | Rosé | Heitz |
| Spain | Prado Enea Gran Reserva | 95 | 79 | Northern Spain | Rioja | Tempranillo Blend | Muga | |
| Spain | Termanthia | 95 | 220 | Northern Spain | Toro | Tinta de Toro | Numanthia | |
| US | Giallo Solare | 95 | 60 | California | Edna Valley | Central Coast | Chardonnay | Center of Effort |
| US | R-Bar-R Ranch | 95 | 45 | California | Santa Cruz Mountains | Central Coast | Pinot Noir | Comartin |
| New Zealand | Maté’s Vineyard | 94 | 57 | Kumeu | Chardonnay | Kumeu River | ||
| US | Shea Vineyard | 94 | 62 | Oregon | Willamette Valley | Pinot Noir | Bergström | |
| US | Abetina | 94 | 105 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi |
| US | Garys’ Vineyard | 94 | 60 | California | Santa Lucia Highlands | Central Coast | Pinot Noir | Roar |
| US | The Funk Estate | 94 | 60 | Washington | Walla Walla Valley (WA) | Columbia Valley | Syrah | Saviah |
| Bulgaria | Bergulé | 90 | 15 | Bulgaria | Mavrud | Villa Melnik | ||
| US | Babushka | 90 | 37 | California | Russian River Valley | Sonoma | Chardonnay | Zepaltas |
| Italy | Vigna Piaggia | 90 | NA | Tuscany | Brunello di Montalcino | Sangiovese | Abbadia Ardenga | |
| France | Nonpareil Trésor Rosé Brut | 90 | 22 | France Other | Vin Mousseux | Sparkling Blend | Bouvet-Ladubay | |
| US | Conner Lee Vineyard | 90 | 42 | Washington | Columbia Valley (WA) | Columbia Valley | Chardonnay | Buty |
| Italy | Riserva | 90 | 135 | Tuscany | Brunello di Montalcino | Sangiovese | Carillon | |
| France | 90 | 60 | Rhône Valley | Châteauneuf-du-Pape | Rhône-style White Blend | Clos de L’Oratoire des Papes | ||
| Italy | 90 | 29 | Tuscany | Vino Nobile di Montepulciano | Sangiovese | Avignonesi | ||
| Italy | 90 | 23 | Tuscany | Chianti Classico | Sangiovese | Casina di Cornia | ||
| Italy | Riserva | 90 | 29 | Tuscany | Chianti Classico | Red Blend | Castello di Monterinaldi | |
| Spain | Amandi | 90 | 17 | Galicia | Ribeira Sacra | Mencía | Don Bernardino | |
| Spain | Alfonso Oloroso Seco | 90 | 26 | Andalucia | Jerez | Palomino | González Byass | |
| US | Private Reserve | 90 | 55 | Idaho | Petite Sirah | Huston | ||
| Italy | Riserva | 90 | 39 | Tuscany | Chianti Classico | Red Blend | Rignana | |
| France | Coteaux | 90 | 69 | Rhône Valley | Cornas | Syrah | Tardieu-Laurent | |
| Italy | Vigneto Odoardo Beccari Riserva | 90 | 30 | Tuscany | Chianti Classico | Red Blend | Vignavecchia | |
| Italy | Poggio alle Mura | 90 | 90 | Tuscany | Brunello di Montalcino | Sangiovese | Banfi | |
| US | Estate Grown | 90 | 60 | California | Mount Veeder | Napa | Cabernet Sauvignon | Brandlin |
| Italy | 90 | 50 | Tuscany | Brunello di Montalcino | Sangiovese | Brunelli Martoccia | ||
| US | 90 | 40 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | Canvasback | |
| Italy | Riserva | 90 | 100 | Tuscany | Brunello di Montalcino | Sangiovese | Capanne Ricci | |
| France | 90 | 68 | Burgundy | Chassagne-Montrachet | Chardonnay | Chartron et Trébuchet | ||
| France | Les 7 Hommes | 90 | 42 | Loire Valley | Sancerre | Sauvignon Blanc | Cherrier Frères | |
| France | L’Inédit | 90 | 28 | Loire Valley | Coteaux du Giennois | Pinot Noir | Clement et Florian Berthier | |
| US | 90 | 18 | California | Russian River Valley | Sonoma | Chardonnay | De Loach | |
| US | Four Flags | 90 | 69 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | DeLille |
| France | Le Pavé | 90 | NA | Loire Valley | Sancerre | Sauvignon Blanc | Domaine Vacheron | |
| US | Reserve | 90 | 25 | New York | Finger Lakes | Finger Lakes | Riesling | Dr. Konstantin Frank |
| US | Final Final | 90 | 30 | Washington | Columbia Valley (WA) | Columbia Valley | Cabernet Sauvignon-Syrah | Efeste |
| Italy | Poggio Bestiale | 90 | 60 | Tuscany | Maremma Toscana | Red Blend | Fattoria di Magliano | |
| Argentina | The Apple Doesn’t Fall Far From The Tree | 91 | 30 | Mendoza Province | Mendoza | Malbec | Matias Riccitelli | |
| Australia | 91 | 36 | Victoria | Mornington Peninsula | Pinot Noir | Moorooduc | ||
| Argentina | Alegoría Gran Reserva | 91 | 25 | Mendoza Province | Mendoza | Malbec | Navarro Correas | |
| France | L’Homme Mort Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Chenevières | |
| Portugal | 91 | 23 | Alentejano | Portuguese Red | Herdade do Rocim | |||
| US | Estate Select | 91 | 36 | California | Santa Clara Valley | Central Coast | Syrah | Jason-Stephens |
| France | Fourchaume Premier Cru | 91 | 38 | Burgundy | Chablis | Chardonnay | Louis Max | |
| US | Animo | 91 | 85 | California | Napa Valley | Napa | Cabernet Sauvignon | Michael Mondavi Family Estate |
| US | Schindler Vineyard | 91 | 50 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Panther Creek |
| US | Barrel Select | 91 | 60 | California | Rutherford | Napa | Cabernet Sauvignon | Provenance Vineyards |
| US | District Collection | 91 | 85 | California | St. Helena | Napa | Cabernet Sauvignon | Raymond |
| US | 91 | 45 | California | Sonoma Coast | Sonoma | Pinot Noir | Red Car | |
| Italy | Bussia Riserva | 91 | NA | Piedmont | Barolo | Nebbiolo | Silvano Bolmida | |
| US | 91 | 19 | Oregon | Willamette Valley | Willamette Valley | Pinot Gris | Trinity Vineyards | |
| Portugal | Premium | 91 | 15 | Alentejo | Portuguese Red | Adega Cooperativa de Borba | ||
| US | Premier Cuvée | 91 | 54 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Archery Summit |
| France | Le Nombre d’Or Brut Nature | 91 | 85 | Champagne | Champagne | Chardonnay | Aubry | |
| US | Juliana Vineyard | 91 | 38 | California | Napa Valley | Napa | Sauvignon Blanc | B Cellars |
| US | 91 | 28 | California | Napa Valley | Napa | Cabernet Sauvignon | B Side | |
| Italy | Boscato | 91 | 75 | Piedmont | Barolo | Nebbiolo | Bel Colle | |
| US | Aeolian | 91 | 42 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Bethel Heights |
| Israel | Reserve | 91 | 25 | Upper Galilee | Cabernet Sauvignon | Binyamina | ||
| Italy | Palliano Riserva | 91 | NA | Piedmont | Roero | Nebbiolo | Ceste | |
| Italy | del Comune di Serralunga d’Alba | 91 | 59 | Piedmont | Barolo | Nebbiolo | Cascina Cucco | |
| Italy | Bricco Luciani | 91 | 85 | Piedmont | Barolo | Nebbiolo | Cascina del Monastero | |
| Italy | Bricco Gattera | 91 | 80 | Piedmont | Barolo | Nebbiolo | Cordero di Montezemolo | |
| France | Montmains Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Gérard Duplessis | |
| US | 91 | 22 | California | Napa Valley | Napa | Cabernet Sauvignon | Eagle Glen | |
| US | Bacigalupi Vineyard | 91 | 65 | California | Russian River Valley | Sonoma | Pinot Noir | Eleven Eleven |
| US | Magnificat | 91 | 50 | California | Napa Valley | Napa | Meritage | Franciscan |
| US | 86 | 10 | California | California | California Other | Cabernet Sauvignon | Belle Ambiance | |
| Portugal | Marquês de Marialva Rosé Bruto | 86 | 12 | Beira Atlantico | Baga | Adega de Cantanhede | ||
| Italy | Nature | 86 | 22 | Veneto | Prosecco | Glera | De Stefani | |
| US | Small Lot Blend | 86 | 13 | California | Mendocino County | Mendocino/Lake Counties | Chardonnay | Parducci |
| Portugal | Muros de Vinha | 86 | 10 | Douro | Portuguese Red | Quinta do Portal | ||
| France | Château Beauvillain-Monpezat | 86 | 14 | Southwest France | Cahors | Malbec-Merlot | Rigal | |
| US | 86 | 18 | California | California | California Other | Chardonnay | The Naked Grape | |
| US | 86 | 36 | California | Sonoma Valley | Sonoma | Cabernet Sauvignon | Tin Barn | |
| France | Pigmentum | 86 | 15 | Southwest France | Buzet | Merlot-Malbec | Georges Vigouroux | |
| France | Pigmentum | 86 | 10 | Southwest France | Côtes de Gascogne | Ugni Blanc-Colombard | Georges Vigouroux |
\[\begin{equation} (x_1,x_2)' \sim \mathcal{N}(\mu, \Sigma) \end{equation}\]
where \(\Sigma\) is dense.
\[\begin{equation} (y_1,y_2)' \sim \mathcal{N}(0, D) \end{equation}\]
where \(D\) is diagonal.
Remember, we’ve assumed:
\[\begin{equation} (x_1,x_2)' \sim \mathcal{N}(\mu, \Sigma) \end{equation}\]
\[\begin{equation} (\tilde x_{1,i}, \tilde x_{2,i}) = (x_{1,i}, x_{2,i}) - (\bar x_{1}, \bar x_{2}) \end{equation}\]
\[\begin{equation} \widehat{\Sigma} = \frac{1}{n}\sum (\tilde x_{1,i}, \tilde x_{2,i})' (\tilde x_{1,i}, \tilde x_{2,i}) \end{equation}\]
\[\begin{equation} \widehat{\Sigma} = P D P' \end{equation}\]
\[\begin{align} y_1 &= P_1' . (\tilde x_1, \tilde x_2)\\ y_2 &= P_2' . (\tilde x_1, \tilde x_2) \end{align}\]
| country | designation | points | price | province | region_1 | region_2 | variety | winery | review_id |
|---|---|---|---|---|---|---|---|---|---|
| US | Martha’s Vineyard | 96 | 235 | California | Napa Valley | Napa | Cabernet Sauvignon | Heitz | 1 |
| Spain | Carodorum Selección Especial Reserva | 96 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | 2 | |
| US | Special Selected Late Harvest | 96 | 90 | California | Knights Valley | Sonoma | Sauvignon Blanc | Macauley | 3 |
| US | Reserve | 96 | 65 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi | 4 |
| France | La Brûlade | 95 | 66 | Provence | Bandol | Provence red blend | Domaine de la Bégude | 5 | |
| Spain | Numanthia | 95 | 73 | Northern Spain | Toro | Tinta de Toro | Numanthia | 6 | |
| Spain | San Román | 95 | 65 | Northern Spain | Toro | Tinta de Toro | Maurodos | 7 | |
| Spain | Carodorum Único Crianza | 95 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | 8 | |
| US | Silice | 95 | 65 | Oregon | Chehalem Mountains | Willamette Valley | Pinot Noir | Bergström | 9 |
| US | Gap’s Crown Vineyard | 95 | 60 | California | Sonoma Coast | Sonoma | Pinot Noir | Blue Farm | 10 |
| Italy | Ronco della Chiesa | 95 | 80 | Northeastern Italy | Collio | Friulano | Borgo del Tiglio | 11 | |
| US | Estate Vineyard Wadensvil Block | 95 | 48 | Oregon | Ribbon Ridge | Willamette Valley | Pinot Noir | Patricia Green Cellars | 12 |
| US | Weber Vineyard | 95 | 48 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Patricia Green Cellars | 13 |
| France | Château Montus Prestige | 95 | 90 | Southwest France | Madiran | Tannat | Vignobles Brumont | 14 | |
| US | Grace Vineyard | 95 | 185 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Domaine Serene | 15 |
| US | Sigrid | 95 | 90 | Oregon | Willamette Valley | Willamette Valley | Chardonnay | Bergström | 16 |
| US | Rainin Vineyard | 95 | 325 | California | Diamond Mountain District | Napa | Cabernet Sauvignon | Hall | 17 |
| Spain | 6 Años Reserva Premium | 95 | 80 | Northern Spain | Ribera del Duero | Tempranillo | Valduero | 18 | |
| France | Le Pigeonnier | 95 | 290 | Southwest France | Cahors | Malbec | Château Lagrézette | 19 | |
| US | Gap’s Crown Vineyard | 95 | 75 | California | Sonoma Coast | Sonoma | Pinot Noir | Gary Farrell | 20 |
| US | Grignolino | 95 | 24 | California | Napa Valley | Napa | Rosé | Heitz | 21 |
| Spain | Prado Enea Gran Reserva | 95 | 79 | Northern Spain | Rioja | Tempranillo Blend | Muga | 22 | |
| Spain | Termanthia | 95 | 220 | Northern Spain | Toro | Tinta de Toro | Numanthia | 23 | |
| US | Giallo Solare | 95 | 60 | California | Edna Valley | Central Coast | Chardonnay | Center of Effort | 24 |
| US | R-Bar-R Ranch | 95 | 45 | California | Santa Cruz Mountains | Central Coast | Pinot Noir | Comartin | 25 |
| New Zealand | Maté’s Vineyard | 94 | 57 | Kumeu | Chardonnay | Kumeu River | 26 | ||
| US | Shea Vineyard | 94 | 62 | Oregon | Willamette Valley | Pinot Noir | Bergström | 27 | |
| US | Abetina | 94 | 105 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi | 28 |
| US | Garys’ Vineyard | 94 | 60 | California | Santa Lucia Highlands | Central Coast | Pinot Noir | Roar | 29 |
| US | The Funk Estate | 94 | 60 | Washington | Walla Walla Valley (WA) | Columbia Valley | Syrah | Saviah | 30 |
| Bulgaria | Bergulé | 90 | 15 | Bulgaria | Mavrud | Villa Melnik | 31 | ||
| US | Babushka | 90 | 37 | California | Russian River Valley | Sonoma | Chardonnay | Zepaltas | 32 |
| Italy | Vigna Piaggia | 90 | NA | Tuscany | Brunello di Montalcino | Sangiovese | Abbadia Ardenga | 33 | |
| France | Nonpareil Trésor Rosé Brut | 90 | 22 | France Other | Vin Mousseux | Sparkling Blend | Bouvet-Ladubay | 34 | |
| US | Conner Lee Vineyard | 90 | 42 | Washington | Columbia Valley (WA) | Columbia Valley | Chardonnay | Buty | 35 |
| Italy | Riserva | 90 | 135 | Tuscany | Brunello di Montalcino | Sangiovese | Carillon | 36 | |
| France | 90 | 60 | Rhône Valley | Châteauneuf-du-Pape | Rhône-style White Blend | Clos de L’Oratoire des Papes | 37 | ||
| Italy | 90 | 29 | Tuscany | Vino Nobile di Montepulciano | Sangiovese | Avignonesi | 38 | ||
| Italy | 90 | 23 | Tuscany | Chianti Classico | Sangiovese | Casina di Cornia | 39 | ||
| Italy | Riserva | 90 | 29 | Tuscany | Chianti Classico | Red Blend | Castello di Monterinaldi | 40 | |
| Spain | Amandi | 90 | 17 | Galicia | Ribeira Sacra | Mencía | Don Bernardino | 41 | |
| Spain | Alfonso Oloroso Seco | 90 | 26 | Andalucia | Jerez | Palomino | González Byass | 42 | |
| US | Private Reserve | 90 | 55 | Idaho | Petite Sirah | Huston | 43 | ||
| Italy | Riserva | 90 | 39 | Tuscany | Chianti Classico | Red Blend | Rignana | 44 | |
| France | Coteaux | 90 | 69 | Rhône Valley | Cornas | Syrah | Tardieu-Laurent | 45 | |
| Italy | Vigneto Odoardo Beccari Riserva | 90 | 30 | Tuscany | Chianti Classico | Red Blend | Vignavecchia | 46 | |
| Italy | Poggio alle Mura | 90 | 90 | Tuscany | Brunello di Montalcino | Sangiovese | Banfi | 47 | |
| US | Estate Grown | 90 | 60 | California | Mount Veeder | Napa | Cabernet Sauvignon | Brandlin | 48 |
| Italy | 90 | 50 | Tuscany | Brunello di Montalcino | Sangiovese | Brunelli Martoccia | 49 | ||
| US | 90 | 40 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | Canvasback | 50 | |
| Italy | Riserva | 90 | 100 | Tuscany | Brunello di Montalcino | Sangiovese | Capanne Ricci | 51 | |
| France | 90 | 68 | Burgundy | Chassagne-Montrachet | Chardonnay | Chartron et Trébuchet | 52 | ||
| France | Les 7 Hommes | 90 | 42 | Loire Valley | Sancerre | Sauvignon Blanc | Cherrier Frères | 53 | |
| France | L’Inédit | 90 | 28 | Loire Valley | Coteaux du Giennois | Pinot Noir | Clement et Florian Berthier | 54 | |
| US | 90 | 18 | California | Russian River Valley | Sonoma | Chardonnay | De Loach | 55 | |
| US | Four Flags | 90 | 69 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | DeLille | 56 |
| France | Le Pavé | 90 | NA | Loire Valley | Sancerre | Sauvignon Blanc | Domaine Vacheron | 57 | |
| US | Reserve | 90 | 25 | New York | Finger Lakes | Finger Lakes | Riesling | Dr. Konstantin Frank | 58 |
| US | Final Final | 90 | 30 | Washington | Columbia Valley (WA) | Columbia Valley | Cabernet Sauvignon-Syrah | Efeste | 59 |
| Italy | Poggio Bestiale | 90 | 60 | Tuscany | Maremma Toscana | Red Blend | Fattoria di Magliano | 60 | |
| Argentina | The Apple Doesn’t Fall Far From The Tree | 91 | 30 | Mendoza Province | Mendoza | Malbec | Matias Riccitelli | 61 | |
| Australia | 91 | 36 | Victoria | Mornington Peninsula | Pinot Noir | Moorooduc | 62 | ||
| Argentina | Alegoría Gran Reserva | 91 | 25 | Mendoza Province | Mendoza | Malbec | Navarro Correas | 63 | |
| France | L’Homme Mort Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Chenevières | 64 | |
| Portugal | 91 | 23 | Alentejano | Portuguese Red | Herdade do Rocim | 65 | |||
| US | Estate Select | 91 | 36 | California | Santa Clara Valley | Central Coast | Syrah | Jason-Stephens | 66 |
| France | Fourchaume Premier Cru | 91 | 38 | Burgundy | Chablis | Chardonnay | Louis Max | 67 | |
| US | Animo | 91 | 85 | California | Napa Valley | Napa | Cabernet Sauvignon | Michael Mondavi Family Estate | 68 |
| US | Schindler Vineyard | 91 | 50 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Panther Creek | 69 |
| US | Barrel Select | 91 | 60 | California | Rutherford | Napa | Cabernet Sauvignon | Provenance Vineyards | 70 |
| US | District Collection | 91 | 85 | California | St. Helena | Napa | Cabernet Sauvignon | Raymond | 71 |
| US | 91 | 45 | California | Sonoma Coast | Sonoma | Pinot Noir | Red Car | 72 | |
| Italy | Bussia Riserva | 91 | NA | Piedmont | Barolo | Nebbiolo | Silvano Bolmida | 73 | |
| US | 91 | 19 | Oregon | Willamette Valley | Willamette Valley | Pinot Gris | Trinity Vineyards | 74 | |
| Portugal | Premium | 91 | 15 | Alentejo | Portuguese Red | Adega Cooperativa de Borba | 75 | ||
| US | Premier Cuvée | 91 | 54 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Archery Summit | 76 |
| France | Le Nombre d’Or Brut Nature | 91 | 85 | Champagne | Champagne | Chardonnay | Aubry | 77 | |
| US | Juliana Vineyard | 91 | 38 | California | Napa Valley | Napa | Sauvignon Blanc | B Cellars | 78 |
| US | 91 | 28 | California | Napa Valley | Napa | Cabernet Sauvignon | B Side | 79 | |
| Italy | Boscato | 91 | 75 | Piedmont | Barolo | Nebbiolo | Bel Colle | 80 | |
| US | Aeolian | 91 | 42 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Bethel Heights | 81 |
| Israel | Reserve | 91 | 25 | Upper Galilee | Cabernet Sauvignon | Binyamina | 82 | ||
| Italy | Palliano Riserva | 91 | NA | Piedmont | Roero | Nebbiolo | Ceste | 83 | |
| Italy | del Comune di Serralunga d’Alba | 91 | 59 | Piedmont | Barolo | Nebbiolo | Cascina Cucco | 84 | |
| Italy | Bricco Luciani | 91 | 85 | Piedmont | Barolo | Nebbiolo | Cascina del Monastero | 85 | |
| Italy | Bricco Gattera | 91 | 80 | Piedmont | Barolo | Nebbiolo | Cordero di Montezemolo | 86 | |
| France | Montmains Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Gérard Duplessis | 87 | |
| US | 91 | 22 | California | Napa Valley | Napa | Cabernet Sauvignon | Eagle Glen | 88 | |
| US | Bacigalupi Vineyard | 91 | 65 | California | Russian River Valley | Sonoma | Pinot Noir | Eleven Eleven | 89 |
| US | Magnificat | 91 | 50 | California | Napa Valley | Napa | Meritage | Franciscan | 90 |
| US | 86 | 10 | California | California | California Other | Cabernet Sauvignon | Belle Ambiance | 91 | |
| Portugal | Marquês de Marialva Rosé Bruto | 86 | 12 | Beira Atlantico | Baga | Adega de Cantanhede | 92 | ||
| Italy | Nature | 86 | 22 | Veneto | Prosecco | Glera | De Stefani | 93 | |
| US | Small Lot Blend | 86 | 13 | California | Mendocino County | Mendocino/Lake Counties | Chardonnay | Parducci | 94 |
| Portugal | Muros de Vinha | 86 | 10 | Douro | Portuguese Red | Quinta do Portal | 95 | ||
| France | Château Beauvillain-Monpezat | 86 | 14 | Southwest France | Cahors | Malbec-Merlot | Rigal | 96 | |
| US | 86 | 18 | California | California | California Other | Chardonnay | The Naked Grape | 97 | |
| US | 86 | 36 | California | Sonoma Valley | Sonoma | Cabernet Sauvignon | Tin Barn | 98 | |
| France | Pigmentum | 86 | 15 | Southwest France | Buzet | Merlot-Malbec | Georges Vigouroux | 99 | |
| France | Pigmentum | 86 | 10 | Southwest France | Côtes de Gascogne | Ugni Blanc-Colombard | Georges Vigouroux | 100 |
| price | points |
|---|---|
| 235 | 96 |
| 110 | 96 |
| 90 | 96 |
| 65 | 96 |
| 66 | 95 |
| 73 | 95 |
| 65 | 95 |
| 110 | 95 |
| 65 | 95 |
| 60 | 95 |
| 80 | 95 |
| 48 | 95 |
| 48 | 95 |
| 90 | 95 |
| 185 | 95 |
| 90 | 95 |
| 325 | 95 |
| 80 | 95 |
| 290 | 95 |
| 75 | 95 |
| 24 | 95 |
| 79 | 95 |
| 220 | 95 |
| 60 | 95 |
| 45 | 95 |
| 57 | 94 |
| 62 | 94 |
| 105 | 94 |
| 60 | 94 |
| 60 | 94 |
| 15 | 90 |
| 37 | 90 |
| NA | 90 |
| 22 | 90 |
| 42 | 90 |
| 135 | 90 |
| 60 | 90 |
| 29 | 90 |
| 23 | 90 |
| 29 | 90 |
| 17 | 90 |
| 26 | 90 |
| 55 | 90 |
| 39 | 90 |
| 69 | 90 |
| 30 | 90 |
| 90 | 90 |
| 60 | 90 |
| 50 | 90 |
| 40 | 90 |
| 100 | 90 |
| 68 | 90 |
| 42 | 90 |
| 28 | 90 |
| 18 | 90 |
| 69 | 90 |
| NA | 90 |
| 25 | 90 |
| 30 | 90 |
| 60 | 90 |
| 30 | 91 |
| 36 | 91 |
| 25 | 91 |
| 45 | 91 |
| 23 | 91 |
| 36 | 91 |
| 38 | 91 |
| 85 | 91 |
| 50 | 91 |
| 60 | 91 |
| 85 | 91 |
| 45 | 91 |
| NA | 91 |
| 19 | 91 |
| 15 | 91 |
| 54 | 91 |
| 85 | 91 |
| 38 | 91 |
| 28 | 91 |
| 75 | 91 |
| 42 | 91 |
| 25 | 91 |
| NA | 91 |
| 59 | 91 |
| 85 | 91 |
| 80 | 91 |
| 45 | 91 |
| 22 | 91 |
| 65 | 91 |
| 50 | 91 |
| 10 | 86 |
| 12 | 86 |
| 22 | 86 |
| 13 | 86 |
| 10 | 86 |
| 14 | 86 |
| 18 | 86 |
| 36 | 86 |
| 15 | 86 |
| 10 | 86 |
Essentially these methods try to learn the structure of the local manifolds in the data. (Also referred to as graph-based or embedding methods.)
\[\begin{equation} p_{j|i} = \frac{\exp(-|x_i-x_j|^2/2\sigma_i^2)}{\sum_{k\neq i} \exp(-|x_i-x_k|^2/2\sigma_i^2)} \end{equation}\]
where \(\sigma_i\) is different for each point: points in dense areas given smaller variance than those in sparse areas
Need a distance metric, so define:
\[\begin{equation} p_{ij} = \frac{1}{N} (p_{j|i} + p_{i|j}) \end{equation}\]
yielding a similarity matrix with elements \(p_{ij}\)
Uses Student-t distribution with one degree of freedom (the Cauchy):
\[\begin{equation} q_{ij} = \frac{1/(1+|z_i-z_j|^2)}{\sum_{k\neq i} 1/(1+|z_i-z_k|^2)} \end{equation}\]
We can calculate the Kullback-Leibler divergence (not a distance!) from \(p_{ij}\rightarrow q_{ij}\):
\[\begin{equation} \text{KL} = \sum_{i,j} p_{ij} \log \frac{p_{ij}}{q_{ij}} \end{equation}\]
Minimise via gradient descent using \(\frac{\partial \text{KL}}{\partial z_i}\)
Suppose
\[\begin{equation} y_i|x_i \sim \mathcal{N}(\alpha + \beta x_i, \sigma), \end{equation}\]
where \((\alpha, \beta, \sigma)\) are parameters to be learned
assume \(\boldsymbol{y}=(y_1,y_2,...,y_K)\) and \(\boldsymbol{x}=(x_1,x_2,...,x_K)\)
\[\begin{equation} p(\alpha,\beta,\sigma|\boldsymbol{y},\boldsymbol{x}) \propto p(\boldsymbol{y}|\boldsymbol{x},\alpha, \beta,\sigma) p(\alpha,\beta,\sigma) \end{equation}\]
use these to determine posterior predictive distribution describing new \(\tilde{\boldsymbol{y}}\) for given (new) \(\tilde{\boldsymbol{x}}\):
\[\begin{equation} p(\tilde{\boldsymbol{y}}|\tilde{\boldsymbol{x}}) = \int p(\tilde{\boldsymbol{y}}|\tilde{\boldsymbol{x}},\alpha, \beta,\sigma) p(\alpha,\beta,\sigma|\boldsymbol{y},\boldsymbol{x}) d\alpha\;d\beta\;d\sigma \end{equation}\]
typically would be done via Markov chain Monte Carlo
\[\begin{equation} y_i = \alpha + \beta x_i + \epsilon_i \end{equation}\]
where \(\epsilon_i\) is an error term. Define mean-squared loss:
\[\begin{equation} L = \frac{1}{K} \sum_{i=1}^{K} (y_i - (\alpha + \beta x_i))^2 \end{equation}\]
determine \(\hat{\alpha}\) and \(\hat{\beta}\) as those minimising \(L\):
\[\begin{align} \frac{\partial L}{\partial \alpha} &= -\frac{2}{K}\sum_{i=1}^{K} (y_i - (\alpha + \beta x_i)) = 0\\ \frac{\partial L}{\partial \beta} &= -\frac{2}{K}\sum_{i=1}^{K} x_i (y_i - (\alpha + \beta x_i)) = 0 \end{align}\]
although a closed form expression exists for \(\hat{\alpha}\) and \(\hat{\beta}\), for more general models, one doesn’t exist \(\implies\) use gradient descent optimisation
\[\begin{align} \alpha &= \alpha - \eta \frac{\partial L}{\partial \alpha}\\ \beta &= \beta - \eta \frac{\partial L}{\partial \beta} \end{align}\]
until \(\alpha\) and \(\beta\) no longer change. \(\eta\) is the learning rate
alternative model:
\[\begin{equation} y_i|x_i \sim \mathcal{N}(f_p(x_i), \sigma), \end{equation}\]
where
\[\begin{equation} f_p(x_i) = \theta_0 + \theta_1 x_i + \theta_2 x_i^2 + ... + \theta_p x_i^p \end{equation}\]
model is better able to fit more complex datasets
\[\begin{equation} L = C||\theta||_q + \frac{1}{K} \sum_{i=1}^{K} (y_i - f_p(x_i))^2 \end{equation}\]
where \(||.||_q\) denotes the \(Lq\) norm: different choices can yield very different estimates
\[\begin{equation} y_i \sim \text{Bernoulli}(\theta_i) \end{equation}\]
where \(0\leq \theta_i \leq 1 = Pr(y_i=1)\)
is given by:
\[\begin{equation} \text{Pr}(y_i|\theta_i) = \theta_i^{y_i} (1 - \theta_i)^{1 - y_i} \end{equation}\]
so that \(\text{Pr}(y_i=1) = \theta_i\) and \(\text{Pr}(y_i=0) = 1 - \theta_i\)
In logistic regression, we use logistic function:
\[\begin{equation} \theta_i = f_\beta(x_i) := \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_i))} \end{equation}\]
assume data are i.i.d., the likelihood is:
\[\begin{equation} L=p(\boldsymbol{y}|\beta,\boldsymbol{x}) = \prod_{i=1}^{K} f_\beta(x_i)^{y_i} (1 - f_\beta(x_i))^{1 - y_i} \end{equation}\]
setting priors, we have a Bayesian model:
\[\begin{equation} p(\beta|\boldsymbol{x}, \boldsymbol{y}) \propto p(\beta) p(\boldsymbol{y}|\beta,\boldsymbol{x}) \end{equation}\]
so can use (say) Markov chain Monte Carlo to estimate model with uncertainty. Frequentists use gradient descent to find maximum likelihood estimates instead
straightforward to extend the model to incorporate multiple regressions:
\[\begin{equation} f_\beta(x_i) := \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}))} \end{equation}\]
But how to interpret parameters of logistic regression?
another way of writing logistic function:
\[\begin{align} f_\beta(x_i) &= \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}))}\\ &= \frac{\exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i})}{1 + \exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i})} \end{align}\]
so that
\[\begin{align} 1 - f_\beta(x_i) = \frac{1}{1 + \exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i})} \end{align}\]
taking the ratio:
\[\begin{equation} \text{odds} = \frac{f_\beta(x_i)}{1-f_\beta(x_i)} = \exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}) \end{equation}\]
so that
\[\begin{equation} \log\text{odds} =\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i} \end{equation}\]
meaning (say) \(\beta_1\) represents the change in log-odds for a one unit change in \(x_{1}\)